On the minimal solution property of neutral functional-differential inclusion
Given a set-valued function \({\mathcal F}\) from \(L^ 1[a,b]\) into subsets of \(L^ 1[a,b]\) the initial-value problem \(\dot x\in {\mathcal F}(x)\), \(x(a)=x_ 0\), is considered. It is proved that if \({\mathcal F}\) is in any sense integral dissipative then a derivative function \(\dot x\) of a solution of the above functional inclusion is a minimal selector for \({\mathcal F}(x)\), i.e., such that \(\| \dot x\| =\min \{\| x\|:\) \(y\in {\mathcal F}(x)\}\). As a corollary of this result a similar theorem for neutral functional-differential inclusions of the form \(\dot x(t)\in F(t,x_ t,\dot x_ t)\) with a real set-valued function F is also obtained.
neutral functional-differential inclusions
scientific article; zbMATH DE number 4204820
scientific article